Structure of the time-optimal control law for multiple arms handling a common object along specified paths

The structure of the time-optimal control law for multiple robot arms cooperatively moving a common object along a specified path with the control torque constraints is addressed. The overall mechanical system is modeled by considering the arms as closed kinematic chains using the Lagrange formulation. This results in a reduced-order dynamic model of the multi-arm system. By parameterizing the resulting dynamic model along a given path, the original higher order optimal control problem with state constraints (path constraints) is transformed into a problem of a double integrator system with state-dependent control constraints that are determined by a linear programming approach. It is then shown that the number of saturated actuators on any finite time interval along the optimal trajectory is [1+3(D-1)(m-1)], where D=2 in 2-dimensional Cartesian space, D=3 in 3-dimensional space, and m is the number of robot arms in the system. The theoretical result is validated by a numerical example